Let’s consider the interference between two sinusoidal sound waves. Two small lo

Question

Let’s consider the interference between two sinusoidal sound waves. Two small loudspeakers A and B (Figure 1) are driven by the same amplifier and each emits a pure sinusoidal wave. The two waves are emitted simultaneously and are initially in step with each other. If the speed of sound is 350 m/s, for what frequencies does maximum constructive interference occur at point P? For what frequencies does destructive interference (cancellation) occur at point P?

The distance from speaker  A to point  P is

[(2.00m)2+(4.00m)2]1/2=4.47m

and the distance from speaker B to point P is

[(1.00m2)+(4.00m2)]1/2=4.12m

The path difference is

d=4.47m4.12m=0.35m

Maximum constructive interference occurs when the path difference d is an integer multiple of :

d=0,v/f,2v/f,...=nv/f(n=0,1,2,...)

So the possible frequencies for maximum constructive interference are

fn==nvd=n350m/s0.35m(n=1,2,3,...)1000Hz,2000Hz,3000Hz,....

Maximum destructive interference (cancellation) occurs when the path difference for the two waves is a half-integer number of wavelengths:

dd==/2,3/2,5/2,...orv/2f,3v/2f,5v/2f,...

The possible frequencies for destructive interference are

fn==nv2d=n350m/s2(0.35m)(n=1,3,5,...)500Hz,1500Hz,2500Hz,....

A)Suppose you move speaker B an additional 2.00 m farther to the right. What is the lowest frequency at which you will hear maximum constructive interference?

B)What is the lowest frequency at which you will hear maximum destructive interference?

Explanation / Answer

A) The distance from speaker A to point P is

[22 + 42]1/2 = 4.47 m

and the distance from speaker B to point P is

[32 + 42]1/2 = 5 m

The path difference is

d = 5 ? 4.47 = 0.53 m

For constructive interference,

d = n? = nc/f

=> f = nc/d

For lowest frequency, n = 1

So, flowest = 1 * 350 / 0.53 = 660.4 Hz

B) For destructive interference,

d = (n + 0.5)? = (n + 0.5)c/f

=> f = (n + 0.5)c/d

For lowest frequency, n = 0

So, flowest = (0 + 0.5) * 350 / 0.53 = 330.2 Hz

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